International Fixed Income
Topic IVA:
International Fixed Income Pricing Hedging
Outline
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•
•
•
Review of foreign bond investments
Hedging and its consequences
Empirical evidence
Concluding remarks
I. Review: Rates of Return on Bonds
Consider a a foreign government bond.
What is it’s US $ rate of return?
P (t  1) S
R (t , t  1) 

P (t )
S
Fn
$
Fn
T 1
Fn
T
$ / Fn
t 1
$ / Fn
t
Taking logs of the above and rearranging gives us
ln[ R (t, t  1)]   ln[ P (t, t  1)]   ln[ S
Fn
$
Fn
T
FN / $
This is approximately equal to:
[yield] - [dur x (r)] - %S(Fn/$)
(t, t  1)]
Rates of Return: Summary
• The $ return on a foreign bond has three
components:
– It’s yield (e.g., coupon, or imputed yield) in the foreign
currency.
– It’s duration component in the foreign currency.
– It’s exchange rate exposure.
• The first two components are always true, while
the second is unique to international fixed
income.
Rates of Return: Summary
Continued...
• The risk associated with this return can be
broken up into two pieces:
– interest rate risk (i.e., duration and maybe convexity)
as the first component (i.e., the coupon) is fixed.
– exchange rate risk.
R
Fn
$
 ( Dur ) 2   2r   2S  2  dur   r ,S
Of course, if there is no exchange rate risk, we just
get the usual result that the volatility of a bond is its
duration times the volatility of rates.
Intuition
• Positive covariance between FN/$ exchange rate and FN interest
rates:
– Interest rates FN/$
- the investor loses money on the bond, and loses
money in the currency market (i.e., minus the exchange rate).
– Why? Suppose inflation in the foreign country goes up, then (I) interest rates rise,
leading to a bond price fall, and (II) the currency depreciates.
• Negative covariance between FN/$ exchange rate and FN interest
rates:
– Interest rates FN/$
- the investor loses money on the bond, but makes
money in the currency market (i.e., minus the exchange rate).
– Why? Suppose monetary policy is tightened, raising interest rates (with
corresponding bond market losses), but attracts foreign capital (with appreciation
in the foreign currency).
II. Hedged Returns
• The previous slides showed unhedged
returns, that is, $-adjusted returns on
foreign bonds, without any hedging of
currency risk.
• In practice, it may be worthwhile hedging
the currency risk using forwards, futures,
or options. Here, we will concentrate on
forwards.
Hedging Mathematics
Recall the unhedged return on $-adjusted foreign bonds:
ln[ R (t, t  1)]   ln[ P (t, t  1)]   ln[ S
Fn
$
Fn
T
FN / $
(t, t  1)]
Two strategies:
(I) Buy foreign bond at price P, and sell all future coupon and
principal payments in exchange for $. (This is like a currency
swap, and transforms the foreign bond into a $ bond via interest
rate parity). We will see that next class.
(II) Alternatively, sell a one-period forward currency contract
equal to next period’s estimated value of the bond (plus interest
if any).
Hedging continued…
If the investor guesses right, and the estimate of next period’s
bond price, PˆT 1 , is close to the true price, PT 1 , then the hedge
will be perfect:
/$
The $ value of his foreign bond will be PˆT 1 Ft ,FN
t 1
Fn
FN
FN / $
FN / $
ˆ
ln[ R (t , t  1)]  ln[ PT 1 / PT ]  ln[ FT ,T 1 / ST ]
Fn
$
The new, hedged return has two components:
(I) the return on the bond in foreign currency terms, just like
before, which includes the yield and capital gain component.
(II) the one-period forward premium we’ve talked about before.
Hedging continued….
• Since the forward premium is known
today, the volatility of the hedged $adjusted return just reflects the volatility of
the bond, i.e., its duration times foreign
interest rate volatility (at least for small
changes).
• What are we assuming here?
Hedging continued….
• We are assuming that the investor perfectly
predicts the price next period. Suppose
this does not occur. Then what?
• Define the prediction error as:
 T 1  P
Fn / $
T 1
Fn / $
ˆ
 PT 1
Hedging continued….
• If  was too large, then our hedge amount
was too small and the unexpected excess
value of the bond is valued at St+1. If  was
too small, then our hedge was too large
and we need to buy additional funds at at
St+1.
• We can therefore measure the return on
the foreign bond in $, hedged but not
perfectly….
Hedging formula….
Fn
FN
FN / $
FN / $
ˆ
ln[ R (t , t  1)]  ln[ PT 1 / PT ]  ln[ FT ,T 1 / ST ]
Fn
$
 ln[  T 1STFN1/ $ / PTFN STFN / $ ]
Three components:
(I) the return from the investor’s predicted price change in
the foreign bond
(II) the forward premium
(III) the unpredicted price change of the foreign bond that is
valued at the FUTURE uncertain exchange rate.
The volatility now involves two parts: (I) the component from
before, and (II) the new unpredictable component which
depends on the volatility of the error, as well as the exchange
rate.
III. Empirical Evidence
• Two cases:
– Monthly data; 10 year-period during 1980’s;
G7 countries, plus global portfolio.
– Weekly data; 1996-1999; G7 countries, as
previous lectures.
• Comparison of unhedged and hedged
returns.
Mean $-adjusted Annualized Return
on Foreign Bond Portfolios (1980’s)
14
12
10
8
Mean - unhedge
Mean - hedge
6
4
Global
US
GER
FRA
CAN
JPN
0
UK
2
Volatility of $-adjusted Annualized Return
on Foreign Bond Portfolios (1980’s)
Global
US
GER
FRA
CAN
JPN
Vol - unhedge
Vol - hedge
UK
20
18
16
14
12
10
8
6
4
2
0
Mean $-adjusted Annualized Return
on Foreign Bond Portfolios (1996-99)
9.00%
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
-1.00%
-2.00%
Mean - unhedge
Mean - hedge
UK JPN CAN FRA GER ITA
US
Vol. $-adjusted Annualized Return
on Foreign Bond Portfolios (1996-99)
14.00%
12.00%
10.00%
8.00%
Vol - unhedge
Vol - hedge
6.00%
4.00%
2.00%
0.00%
UK
JPN CAN FRA GER ITA
US
Recall: % of Volatility of $-adjusted
Foreign Bond Due to Currency Risk
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Dur.=10
Dur.=5
Dur.=1
UK
JPN
CAN
FRA
GER
ITA
Concluding Remarks….
• Is hedging having your cake and eating it too?
– As a long-run matter, the interest rate differential should offset
exchange rates, so passive strategies should produce the same
level of returns.
– Here, most of the correlations between interest rates and
exchange rates moved in the opposite direction of this theory
(except Canada), so their were ex-post gains to hedging.
• Alternatively, what we have shown is that hedged $adjusted foreign bond returns subjects you to foreign
interest rate movements.
– These foreign-based movements depend on real economic and
central bank risk in those countries. By diversifying across
countries in $ terms, you can reduce your overall exposure to
country-specific risks.